Single-Flighted Screw Elements With a Reduced Tip Angle

ABSTRACT

The invention relates to screw elements for multi-screw extruders with screws co-rotating in pairs and being fully self-wiping in pairs, to the use of the screw elements in multi-screw extruders and to a process for extruding plastic compositions.

This is an application filed under 35 USC §371 of PCT/EP2009/004251,claiming priority to DE 10 2008 029 303.0 filed on Jun. 20, 2009.

BACKGROUND OF THE INVENTION

(1) Field of the Invention

The invention relates to screw elements for multi-screw extruders withscrews co-rotating in pairs and being fully self-wiping in pairs, to theuse of the screw elements in multi-screw extruders and to a process forextruding plastic compositions.

(2) Description of Related Art

Co-rotating twin- or optionally multi-screw extruders, the rotors ofwhich are fully self-wiping, have long been known (see for example DP862 668). Extruders which are based on the principle of fullyself-wiping profiles have been put to many different uses in polymerproduction and processing. This is primarily a consequence of the factthat polymer melts adhere to surfaces and degrade over time atconventional processing temperatures, which is prevented by theself-cleaning action of the fully self-wiping screws. Rules forproducing fully self-wiping screw profiles are stated, for example, inKlemens Kohlgrüber: Der gleichläufige Doppelschneckenextruder [Theco-rotating twin-screw extruder], Hanser Verlag Munich 2007, p. 96 etseq.). It is also described therein how a predetermined screw profile ofthe 1st screw of a twin-screw extruder determines the screw profile ofthe 2nd screw of a twin-screw extruder. The screw profile of the 1stscrew of the twin-screw extruder is therefore known as the generatingscrew profile. The screw profile of the 2nd screw of the twin-screwextruder follows from the screw profile of the 1st screw of thetwin-screw extruder and is therefore known as the generated screwprofile. In the case of a multi-screw extruder, neighbouring screws arealways arranged alternately with a generating screw profile and agenerated screw profile.

Modern twin-screw extruders have a building-block system, in whichvarious screw elements may be mounted on a core shaft. In this way, aperson skilled in the art may adapt the twin-screw extruder to theparticular task in hand.

A plastic composition is taken to mean a deformable composition.Examples of plastic compositions are polymer melts, especially ofthermoplastics, and elastomers, mixtures of polymer melts or dispersionsof polymer melts with solids, liquids or gases.

The extrusion of plastic compositions plays a major role in particularin the production, compounding and processing of polymers. Extrusion istaken to mean the treatment of a substance or mixture of substances in aco-rotating twin- or multi-screw extruder, as is comprehensivelydescribed in Kohlgrüber.

During polymer production, extrusion is performed, for example, to degasthe polymers (see for example Kohlgrüber pages 191 to 212).

During polymer compounding, extrusion is performed, for example, toincorporate additives or to mix various polymers which differ, forexample, in chemical composition, molecular weight or molecularstructure (see for example Kohlgrüber pages 59 to 93). Compoundinginvolves the conversion of a polymer into a finished plastics mouldingcomposition (or compound) using plastics raw materials, which areconventionally plasticized, and adding and incorporating fillers and/orreinforcing materials, plasticizers, bonding agents, slip agents,stabilizers, colours etc. Compounding often also includes the removal ofvolatile constituents such as for example air and water. Compounding mayalso involve a chemical reaction such as for example grafting,modification of functional groups or molecular weight modifications bydeliberately increasing or decreasing molecular weight.

During polymer processing, the polymers are preferably converted intothe form of a semi-finished product, a ready-to-use product or acomponent. Processing may proceed, for example, by injection moulding,extrusion, film blowing, calendering or spinning Processing may alsoinvolve mixing polymers with fillers and auxiliary substances andadditives as well as chemical modifications such as for examplevulcanization.

The treatment of plastic compositions during extrusion includes one ormore of the operations: conveying, melting, dispersion, mixing,degassing and pressure build-up.

As is generally known and described, for example, in Kohlgrüber on pages169 to 190, mixing may be differentiated into distributive anddispersive mixing. Distributive mixing is taken to mean the uniformdistribution of various components in a given volume. Distributivemixing occurs, for example, when similar or mutually compatible polymersare mixed. In dispersive mixing, solid particles, fluid droplets or gasbubbles are firstly subdivided. Subdivision entails applyingsufficiently large shear forces in order, for example, to overcome thesurface tension at the interface between the polymer melt and anadditive.

Melt conveying and pressure build-up are described on pages 73 et seq.of publication Kohlgrüber. The melt conveying zones in extruder screwsserve to transport the product from one processing zone to the next andto draw in fillers. Melt conveying zones are generally partially filled,such as for example during the transport of the product from oneprocessing zone to the next, during degassing and in holding zones. Theenergy required for conveying is dissipated and is disadvantageouslymanifested by an increase in the temperature of the polymer melt. Thescrew elements used in a conveying zone should therefore be those whichdissipate the least possible energy. Thread elements having pitches of1× the extruder barrel diameter D are conventional Kohlgrüber for simplemelt conveying.

Upstream of pressure consumers within the extruder, such as for examplebackward conveying elements, mixing elements, backward conveying orneutral kneading blocks, and upstream of pressure consumers outside theextruder, such as for example die plates, extrusion dies and meltfilters there is formed a back pressure zone within the extruder, inwhich conveying is carried out in a completely full state and in whichthe pressure for overcoming the pressure consumer must be built up. Thepressure build-up zone of an extruder, in which the pressure required tooutput the melt is generated, is known as the metering zone. The energyintroduced into the polymer melt is divided into effective power forpressure build-up and for conveying the melt and dissipation power whichis disadvantageously manifested by an increase in the temperature of themelt. In the pressure build-up zone, strong reflux of the melt occursover the screw tips, so resulting in elevated energy input Kohlgrüber.Screw elements used in a pressure build-up zone should therefore bethose which dissipate the least possible energy.

A person skilled in the art furthermore knows (Kohlgrüber, pages 129 to146) that efficiency during pressure build-up of double-flightedconveying elements with the known Erdmenger screw profile is around 10%.A pressure rise of 50 bar at a melt density of 1000 kg/m³ and a thermalcapacity of the melt of 2000 J/kg/K results at said efficiency of 10% ina temperature rise of 25 K (Kohlgrüber, page 120). This heating mayresult in harm to the product such as for example a change in odour,colour, chemical composition or molecular weight or in the formation ofnon-uniformities in the product such as gel particles or specks.

When extruding polyethylene and polyethylene copolymers, an excessivelyhigh temperature results in an increase in molecular weight, branchingand crosslinking. Polyethylene and polyethylene copolymers furthermorereact with atmospheric oxygen in the autoxidation cycle known to aperson skilled in the art (Hepperle, J.: Schädigungsmechanismen beiPolymeren [Damage mechanisms in polymers], Polymeraufbereitung 2002,VDI-K, VDI-Verlag GmbH, Zweifel, H.: Stabilization of PolymericMaterials. Springer, Berlin, 1997, Schwarzenbach, K. et al.:Antioxidants, in Zweifel, H. (ed.): Plastics Additives Handbook, Hanser,Munich, 2001, Cheng, H. N., Schilling, F. C., Bovey, F. A.: 13C NuclearMagnetic Resonance Observation of the Oxidation of Polyethylene,Macromolecules 9 (1976) p. 363-365) to form strong-smelling and thusdisruptive low molecular weight components such as for example ketones,aldehydes, carboxylic acids, hydroperoxides, esters, lactones andalcohols.

When extruding copolymers based on polyethylene and vinyl acetate, anexcessively high temperature additionally results in the formation ofstrong-smelling and corrosive acetic acid.

When extruding polypropylene and polypropylene copolymers, anexcessively high temperature results in molecular weight degradation.Polypropylene and polypropylene copolymers furthermore react withatmospheric oxygen in the autoxidation cycle to form strong-smelling andthus disruptive low molecular weight components such as for exampleketones, aldehydes, carboxylic acids, hydroperoxides, esters, lactonesand alcohols.

When extruding polyvinyl chloride, an excessively high temperatureresults in product discoloration and the elimination of corrosivegaseous hydrochloric acid, wherein the hydrochloric acid in turncatalyses further elimination of hydrochloric acid.

When extruding polystyrene, an excessively high temperature results inthe formation of harmful styrene as well as dimeric and trimericstyrene, with molecular weight degradation and corresponding impairmentof mechanical properties.

When extruding polystyrene-acrylonitrile copolymer (SAN), the productturns a yellowish colour on exposure to thermal stress, resulting inreduced transparency, and forms the carcinogenic monomer acrylonitrileas well as styrene, with molecular weight degradation and impairment ofmechanical properties.

When extruding aromatic polycarbonates, the product turns a yellowishcolour on exposure to excessive thermal stress, in particular due to theaction of oxygen, resulting in reduced transparency, and exhibitsmolecular weight degradation, in particular due to the action of water.Monomers such as for example bisphenol A are also dissociated onexposure to elevated temperature.

When extruding polyesters such as for example polyethyleneterephthalate, polybutylene terephthalate and polytrimethyleneterephthalate, an excessive temperature and the action of water resultin a reduction in molecular weight and displacement of the end groups inthe molecule. This is problematic especially when recycling polyethyleneterephthalate. Polyethylene terephthalate eliminates acetaldehyde atelevated temperature, which may result in changes to the flavour of thecontents of beverage bottles, for example.

When extruding thermoplastics impact-modified with diene rubbers, inparticular with butadiene rubber, in particular impact-modified gradesof polystyrene (HIPS) and impact-modified SAN(acrylonitrile-butadiene-styrene, ABS), an excessive temperature resultsin the elimination of carcinogenic butadiene and acrylonitrile and toxicvinylcyclohexene. Furthermore the diene rubber crosslinks, resulting inimpaired mechanical properties of the product.

When extruding polyoxymethylene, an excessive temperature results in theelimination of toxic formaldehyde.

When extruding polyamides such as polyamide 6, polyamide 6,6, polyamide4,6, polyamide 11 and polyamide 12, an excessively high temperatureresults in product discoloration and molecular weight degradation and inthe reformation of monomers and dimers, so resulting in impairment ofmechanical properties, especially in the presence of water.

When extruding thermoplastic polyurethanes, an excessively hightemperature results in changes to the molecular structure bytransurethanization and, in the presence of water, in molecular weightdegradation. Both of these undesirably influence the properties of thethermoplastic polyurethane.

When extruding polymethyl methacrylate, methyl methacrylate iseliminated and molecular weight degraded on exposure to excessivethermal stress, resulting in an odour nuisance and impaired mechanicalproperties.

When extruding polyphenylene sulphide, an excessively high temperatureresults in the elimination of sulphur-containing organic and inorganiccompounds, which result in an odour nuisance and may lead to corrosionon the extrusion dies. Low molecular weight oligomers and monomers arealso formed and the molecular weight degraded, so impairing themechanical properties of polyphenylene sulphide.

When extruding polyphenylsulphone, an excessively high temperatureresults in the elimination of organic compounds, especially in thepresence of water. The molecular weight also declines, resulting inimpaired mechanical properties.

When extruding polyphenylene ether, excessively high temperatures resultin the elimination of low molecular weight organic compounds, whereinthe molecular weight declines. This results in impairment of themechanical properties of the product.

When extruding diene rubbers such as for example polybutadiene (BR),natural rubber (NR) and synthetic polyisoprene (IR), butyl rubber (IIR),chlorobutyl rubber (CIIR), bromobutyl rubber (BIIR), styrene-butadienerubber (SBR), polychloroprene (CR), butadiene-acrylonitrile rubber(NBR), partially hydrogenated butadiene-acrylonitrile rubber (HNBR) andethylene-propylene-diene copolymers (EPDM), an excessively hightemperature results in gel formation by crosslinking, which leads to theimpairment of mechanical properties of components produced therefrom. Inthe case of chloro- and bromobutyl rubber, an elevated temperature mayresult in the elimination of corrosive gaseous hydrochloric orhydrobromic acid, which in turn catalyses further decomposition of thepolymer.

Excessively high temperatures during extrusion result in prematurevulcanization of rubber compounds which contain vulcanizing agents, suchas for example sulphur or peroxides. This results in its no longer beingpossible to produce any products from these rubber compounds.

When extruding mixtures of one or more polymers at excessively hightemperatures, the disadvantages of extruding the individual polymersoccur in each case.

The energy input into a twin-screw extruder is determined by the processparameters throughput and rotational speed, by the material propertiesof the product and by the geometry of the screws used.

According to the prior art Kohlgrüber (see for example page 101), thegeometry of fully self-wiping screw elements is defined by stating theindependent variables number of flights z, centreline distance a andouter radius ra of the fully self-wiping contour. According to the priorart, the tip angle, over which all points of the profile clean thebarrel, is not a variable which is adjustable and adaptable to theproblem addressed, but is instead obtained from Eq. 1

$\begin{matrix}{{{KW}\; 0} = {\frac{\pi}{z} - {2{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}}} & \left( {{Eq}.\mspace{14mu} 1} \right)\end{matrix}$

wherein KW0 is the tip angle of the fully self-wiping profile in radiansand π the circle constant (π≈3.14159).

According to the prior art, the sum of the tip angles over both elementsof a closely intermeshing pair of elements SKW0 is inevitably obtainedfrom

$\begin{matrix}{{{SKW}\; 0} = {{2\pi} - {4z\; {\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}}} & \left( {{Eq}.\mspace{14mu} 2} \right)\end{matrix}$

Screw profiles may be constructed with one or more screw flights. Knownscrew profiles with exactly one screw flight are known for goodconveying capacity and rigidity during pressure build-up. They have avery wide screw tip which cleans the screw barrel with a narrow gap. Itis known to a person skilled in the art that in the region of the screwtips, due to the narrow gap, a particularly large amount of energy isdissipated in the melt, which leads locally to significant overheatingin the product. A large tip angle, in particular, is harmful in thisrespect.

It is therefore double-flighted screw profiles having only a narrowscrew tip which are predominantly used in co-rotating twin-screwextruders according to the prior art. However, these are considerablyless effective in pressure build-up than are single-flighted screwprofiles.

Drive energy is supplied to the twin-screw extruder in the form ofhigh-grade electrical energy, such that a reduction in energy input isalso desirable on cost and environmental grounds. Furthermore, in manyprocesses a high energy input also limits the possible throughput of thetwin-screw extruder and thus its economic viability.

U.S. Pat. No. 3,900,187 describes single-flighted screw profiles with areduced tip angle. If they have sufficiently effective pressurebuild-up, screw elements with such screw profiles have a lower shearingaction than other known single-flighted screw elements. U.S. Pat. No.3,900,187, however, merely discloses the production of axiallysymmetrical screw profiles, in which the profile flank zone adjoiningthe screw tip is represented by a circular arc, the centre point ofwhich lies on the perpendicular to the axis of symmetry of the profilethrough the point of rotation. The in U.S. Pat. No. 3,900,187 thuscannot be individually adapted to specific problems and are limited intheir application.

On the basis of the prior art, the object is therefore to provide screwelements which produce pressure build-up comparable to knownsingle-flighted screw elements, but which apply less shear to thematerial to be processed and thus do not impair product quality. Itshould be possible to construct the desired screw elements in asversatile a manner as possible in order to be able to adapt the shearand strain stresses applied by the rotating screw profiles to thepolymers to be processed to the particular problem addressed.

Screw elements have surprisingly been found with which this object maybe achieved and in which the sum of the tip angles of a pair of elementsis less than

${{2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}},$

wherein in the case of axially symmetrical screw profiles none of thecentre points of the flank circles lies on the perpendicular to the axisof symmetry of the profile, which axis of symmetry passes through thepoint of rotation.

BRIEF SUMMARY OF THE INVENTION

The present invention accordingly provides screw elements formulti-screw extruders with screws co-rotating in pairs and being fullyself-wiping in pairs, with in each case exactly one screw flight, with acentreline distance a and outer radius ra, characterized in that the sumof the tip angles of a generating and of a generated screw profile isless than

${2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}$

and, in the case of axially symmetrical screw profiles, none of thecentre points of the flank circles lies on the perpendicular to the axisof symmetry of the profile, which axis of symmetry passes through thepoint of rotation.

The cross-sectional profiles, hereinafter also known for short asprofiles or also screw profiles, of screw elements according to theinvention may be unambiguously described by an arrangement of circulararcs.

The screw profiles of screw elements according to the invention arepreferably composed in cross-section of n circular arcs, wherein n is aninteger greater than 4. Each of the n circular arcs has a starting andan end point.

The position of each circular arc j (j=1 to n) may be unambiguouslyestablished by stating two different points. The position of a circulararc is conveniently established by stating the centre point and thestarting or end point. The magnitude of an individual circular arc j isestablished by the radius r_(j) and the angle α_(j) about the centrepoint between the starting and end point, wherein the radius r_(j) isgreater than or equal to 0 and less than or equal to the centrelinedistance a between the screws and the angle α_(j) in radians is greaterthan or equal to 0 and less than or equal to 2π, wherein π is the circleconstant (π≈3.14159).

The profiles of screw elements according to the invention may alsocomprise one or more “kinks” A kink is conveniently treated as acircular arc with a radius r=0. The “magnitude of the kink” isdetermined by the corresponding angle of the circular arc with theradius r=0, i.e. at a kink there is a transition from a first circulararc by rotation about the angle of a second circular arc with a radiusr=0 to a third circular arc. Or in other words: a tangent to the firstcircular arc in the centre point of the second circular arc with theradius r=0 intersects a tangent to the third circular arc likewise inthe centre point of the second circular arc at an angle whichcorresponds to the angle of the second circular arc. Taking account ofthe second circular arc, all adjacent circular arcs (first→second→third)merge tangentially into one another. A circular arc with a radius r=0 isconveniently treated as a circular arc whose radius is equal to eps,wherein eps is a very small positive real number which tends towards 0(eps<<1, eps→0).

In a profile according to the invention, the circular arcs always mergetangentially into one another at their start and end points.

The zones of a screw profile which are equal to the outer screw radiusare known as tip zones. The zones of a screw profile which are equal tothe core radius are known as grooved zones. The zones of a screw profilewhich are smaller than the outer screw radius and larger than the coreradius are known as flank zones.

Screw elements according to the invention are characterized in theircross-section in that

-   -   the generating screw profile and the generated screw profile lie        in one plane,    -   the axis of rotation of the generating screw profile and the        axis of rotation of the generated screw profile at a distance a        (centreline distance) are in each case perpendicular to said        plane of the screw profiles, the point of intersection of the        axis of rotation of the generating screw profile with said plane        being designated as the point of rotation of the generating        screw profile and the point of intersection of the axis of        rotation of the generated screw profile with said plane being        designated as the point of rotation of the generated screw        profile,    -   the number of the circular arcs of the generating screw profile        is n,    -   the outer radius ra of the generating screw profile is greater        than or equal to 0 (ra≧0) and less than or equal to the        centreline distance (ra≦a),    -   the core radius ri of the generating screw profile is greater        than 0 (ri>0) and less than or equal to ra (ri≦ra),    -   the circular arcs of the generating screw profile form a closed        profile, i.e. the sum of the angles α_(j) of all the circular        arcs j is equal to 2π,    -   the circular arcs of the generating screw profile form a convex        profile,    -   each of the circular arcs of the generating screw profile lies        within or at the limits of a circular ring with the outer radius        ra and the core radius ri, the centre point of which lies on the        point of rotation of the generating screw profile,    -   exactly one of the circular arcs of the generating screw profile        has the outer radius ra of the generating screw profile,    -   exactly one of the circular arcs of the generating screw profile        has the core radius ri of the generating screw profile,    -   the number of circular arcs n′ of the generated screw profile is        equal to the number of circular arcs n of the generating screw        profile,    -   the outer radius ra′ of the generated screw profile is equal to        the difference between the centreline distance and core radius        ri of the generating screw profile (ra′=a−ri),    -   the core radius ri′ of the generated screw profile is equal to        the difference between the centreline distance and outer radius        ra of the generating screw profile (ri′=a−ra),    -   the angle α_(j)′ of the j′th circular arc of the generated screw        profile is equal to the angle α_(j) of the jth circular arc of        the generating screw profile, j and j′ being integers which pass        jointly through all the values in the range from 1 to the number        of circular arcs n or n′ respectively,    -   the sum of radius r_(j)′ of the j′th circular arc of the        generated screw profile and radius r_(j) of the jth circular arc        of the generating screw profile is equal to the centreline        distance a, j and j′ being integers which pass jointly through        all the values in the range from 1 to the number of circular        arcs n or n′ respectively,    -   the centre point of the j′th circular arc of the generated screw        profile is at a distance from the centre point of the jth        circular arc of the generating screw profile which is equal to        the centreline distance a, and the centre point of the j′th        circular arc of the generated screw profile is at a distance        from the point of rotation of the generated screw profile which        is equal to the distance of the centre point of the jth circular        arc of the generating screw profile from the point of rotation        of the generating screw profile, and the connecting line between        the centre point of the j′th circular arc of the generated screw        profile and the centre point of the jth circular arc of the        generating screw profile is a line parallel to a connecting line        between the point of rotation of the generated screw profile and        the point of rotation of the generating screw profile, j and j′        being integers which pass jointly through all the values in the        range from 1 to the number of circular arcs n or n′        respectively,    -   a starting point of the j′th circular arc of the generated screw        profile lies in a direction relative to the centre point of the        j′th circular arc of the generated screw profile which is        opposite to that direction which a starting point of the jth        circular arc of the generating screw profile has relative to the        centre point of the jth circular arc of the generating screw        profile, j and j′ being integers which pass jointly through all        the values in the range from 1 to the number of circular arcs n        or n′ respectively,    -   the sum of the tip angles of a generating and of a generated        screw profile is less than

${{2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}},$

and

-   -   in the case of axially symmetrical screw profiles none of the        centre points of the flank circles lies on the perpendicular to        the axis of symmetry of the profile, which axis of symmetry        passes through the point of rotation.

The sum of the tip angles of a generating and of a generated screwprofile is less than

${{2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}},$

preferably less than

${0.8 \cdot \left( {{2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}} \right)},$

particularly preferably less than

${0.6 \cdot \left( {{2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}} \right)},$

and most preferably less than

$0.4 \cdot {\left( {{2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}} \right).}$

In screw elements according to the invention in each case onecross-sectional profile is preferably composed of five or more circulararcs with a radius greater than or equal to zero and less than or equalto a, the circular arcs merging tangentially into one another at theirend points.

The profiles of screw elements according to the invention may in eachcase be asymmetrical or symmetrical relative to an axis through thepoint of rotation of the respective screw element. Axially symmetricalprofiles of screw elements according to the invention are distinguishedin that none of the centre points of the circular arcs which form theflank zone lies on the perpendicular to the axis of symmetry of theprofile, which axis of symmetry passes through the point of rotation.

The profiles of screw elements according to the invention aredistinguished in that they may be designed solely using a set square andpair of compasses. The tangential transition between the jth and the(j+1)th circular arc of the generating screw profile is thus designed bydescribing a circle with the radius r_(j+1) about the end point of thejth circular arc, and the point of intersection, located closer to thepoint of rotation of the generating screw profile, of this circle withthe straight line which is defined by the centre point and the end pointof the jth circular arc is the centre point of the (j+1)th circular arc.

It is recommended that the method for producing screw profiles becarried out on a computer. The dimensions of the screw elements are thenpresent in a form in which they may be supplied to a CAD milling machinefor producing the screw elements.

The outer screw radius normalized to the centreline distance of screwelements according to the invention is preferably in the range from 0.51to 0.7, particularly preferably in the range from 0.52 to 0.66 and veryparticularly preferably in the range from 0.57 to 0.63.

The screw elements according to the invention may be constructed asconveying elements or kneading elements or mixing elements.

A conveying element is known to be distinguished in that (see forexample Kohlgrüber, pages 227-248) the screw profile is rotated andextended continuously helically in the axial direction. Depending on thedirection of rotation of the screws, the conveying element is of right-or left-handed construction. Backward conveying elements are in eachcase obtained by a contrary twist. The pitch of the conveying element ispreferably in the range from 0.1 to 10 times the centreline distance,the pitch being taken to mean the axial length which is necessary forone complete rotation of the screw profile, and the axial length of aconveying element is preferably in the range from 0.1 to 10 times thescrew diameter.

A kneading element is known to be distinguished in that (see for exampleKohlgrüber, pages 227-248) the screw profile extends discontinuously inthe axial direction in the form of kneading discs. The kneading discsmay be arranged in right- or left-handed manner or neutrally. The axiallength of the kneading discs is preferably in the range from 0.05 to 10times the centre distance. The axial distance between two adjacentkneading discs is preferably in the range from 0.002 to 0.1 times thescrew diameter.

As is known, mixing elements are formed (see for example, Kohlgrüberpages 227-248) by constructing conveying elements with openings in thescrew tips. The mixing elements may be right- or left-handed. Theirpitch is preferably in the range from 0.1 to 10 times the centrelinedistance and the axial length of the elements is preferably in the rangefrom 0.1 times to 10 times the centreline distance. The openingspreferably take the form a U- or V-shaped groove, which is preferablyarranged in a counter-conveying or axially parallel manner.

Once they have been designed, preferably on a computer, taking accountof the above-stated design features, the screw elements according to theinvention may be produced for example using a milling machine. Preferredmaterials for producing the screw elements are steels, in particularnitriding steels, chromium, tool and special steels, as well as metalliccomposite materials based on iron, nickel or cobalt and produced bypowder metallurgy.

A person skilled in the art is aware that fully self-wiping screwprofiles cannot be used directly in a twin-screw extruder. Instead,clearances are required between the screws. To this end, variousstrategies are described in Kohlgrüber on page 28 et seq. For screwprofiles of screw elements according to the invention, clearances in therange from 0.001 to 0.1, relative to the diameter of the screw profile,are used, preferably from 0.002 to 0.05 and particularly preferably from0.004 to 0.02. The clearances may, as is known to a person skilled inthe art, be of different dimensions or identical between screw andbarrel and between screw and screw. The clearances may also be constantor, within the stated limits, variable. It is also possible to move ascrew profile within the clearances. Possible clearance strategies arethe possibilities, described in on page 28 et seq., of centrelinedistance enlargement, longitudinal section offsets and three-dimensionaloffsets, all of which are known to a person skilled in the art. In thecase of centreline distance enlargement, a screw profile of a relativelysmall diameter is constructed and spaced further apart by the amount ofclearance between the screws. In the longitudinal section offset method,the longitudinal section profile curve (parallel to the axis) isdisplaced inwards by half the screw-screw clearance. In thethree-dimensional offset method, starting from the three-dimensionalcurve on which the screw elements clean one another, the screw elementis reduced in size in the direction perpendicular to the faces of thefully self-wiping profile by half the clearance between screw and screw.The longitudinal section and three-dimensional offset methods arepreferred, the three-dimensional offset method being particularlypreferred.

The profiles of screw elements according to the invention may beconstructed using a method described in PCT/EP2009/003549.

The present invention further provides use of the screw elementsaccording to the invention in multi-screw extruders. The screw elementsaccording to the invention are preferably used in twin-screw extruders.The screw elements may be present in the multi-screw extruders in theform of kneading, mixing or conveying elements. It is likewise possibleto combine kneading, conveying and mixing elements with one another inone extruder. The screw elements according to the invention may also becombined with other screw elements, which are for example knownaccording to the prior art.

The present invention further provides a process for extruding plasticcompositions in a twin-screw or multi-screw extruder using screwelements according to the invention, characterized in that the sum ofthe tip angles of a pair of screw elements is less than

${2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}$

and, in the case of axially symmetrical screw profiles, none of thecentre points of the flank circles lies on the perpendicular to the axisof symmetry of the profile, which axis of symmetry passes through thepoint of rotation.

Plastic compositions which may be extruded highly efficiently accordingto the invention while gentle treatment of the product is simultaneouslyensured, are for example suspensions, pastes, glass, ceramiccompositions, metals in the form of a melt, plastics, plastics melts,polymer solutions, elastomer and rubber compositions.

Plastics and polymer solutions are preferably used, particularlypreferably thermoplastic polymers. Preferred thermoplastic polymers arepreferably at least one of the series of polycarbonate, polyamide,polyester, in particular polybutylene terephthalate and polyethyleneterephthalate, and polyether, thermoplastic polyurethane, polyacetal,fluoropolymer, in particular polyvinylidene fluoride, and polyethersulphones, polyolefin, in particular polyethylene and polypropylene, andpolyimide, polyacrylate, in particular poly(methyl)methacrylate, andpolyphenylene oxide, polyphenylene sulphide, polyether ketone,polyarylether ketone, styrene polymers, in particular polystyrene, andstyrene copolymers, in particular styrene-acrylonitrile copolymer,acrylonitrile-butadiene-styrene block copolymers and polyvinyl chloride.Blends of the listed plastics are likewise preferably used, these beingunderstood by a person skilled in the art to be a combination of two ormore plastics.

Further preferred feed materials are elastomers. Preferred elastomersare preferably at least one from the series of styrene-butadiene rubber,natural rubber, butadiene rubber, isoprene rubber,ethylene-propylene-diene rubber, ethylene-propylene rubber,butadiene-acrylonitrile rubber, hydrogenated nitrile rubber, butylrubber, halobutyl rubber, chloroprene rubber, ethylene-vinyl acetaterubber, polyurethane rubber, thermoplastic polyurethane, gutta percha,acrylate rubber, fluororubber, silicone rubber, sulphide rubber,chlorosulphonyl-polyethylene rubber. A combination of two or more of thelisted rubbers, or a combination of one or more rubbers with one or moreplastics is of course also possible.

These thermoplastics and elastomers may be used in pure form or asmixtures with fillers and reinforcing materials, such as in particularglass fibres, as mixtures with one another or with other polymers or asmixtures with conventional polymer additives.

In one preferred embodiment the plastics compositions, in particular thepolymer melts and mixtures of polymer melts, have additives admixed withthem. These may be placed as solids, liquids or solutions in theextruder together with the polymer or instead at least some of theadditives or all the additives are supplied to the extruder via a sidestream.

Additives may impart many different characteristics to a polymer. Theymay for example be colorants, pigments, processing auxiliaries, fillers,antioxidants, reinforcing materials, UV absorbers and light stabilizers,metal deactivators, peroxide scavengers, basic stabilizers, nucleatingagents, benzofurans and indolinones active as stabilizers orantioxidants, mould release agents, flame-retardant additives,antistatic agents, dye preparations and melt stabilizers. Examples offillers and reinforcing materials are carbon black, glass fibres, clay,mica, graphite fibres, titanium dioxide, carbon fibres, carbonnanotubes, ionic liquids and natural fibres.

The invention is explained in greater detail below by way of examplewith reference to the figures without however being restricted thereto.

The following nomenclature is used in the figures.

-   -   All dimensions are normalized to the centreline distance a. The        normalized dimensions are denoted with capital letters. Example:        normalized outer radius: RA=ra/a.    -   Angles are stated in radians.    -   Mx and My are the x and y coordinates of the circle centre point        of a profile-generating circular arc in a Cartesian system of        coordinates, the origin of which is located at the point of        rotation of the screw profile.    -   The circular arc with the radius r=ra is denoted “1”. It defines        the contour of the screw tip.    -   The circular arc with the radius r=ri is denoted “1′”. It        defines the contour of the grooved zone of the screw profile.    -   The circular arcs “2” and “2′”, “3” and “3′” etc. define the        flank of the screw profile.    -   R is the radius normalized to the centreline distance a and α        the arc angle of the circular arc.    -   Further abbreviations: RG=normalized barrel radius,        RV=normalized virtual barrel radius, RA=normalized outer radius        of the fully self-wiping profile, RF=normalized outer radius of        the screw to be manufactured, S=normalized clearance of the        screws relative to one another (gap), D=normalized clearance of        screw to barrel, VPR=normalized amount of profile displacement,        VPW angle of profile displacement in radians, VLR=normalized        amount of left-hand screw displacement, VLW=angle of left-hand        screw displacement, VRR=normalized amount of right-hand screw        displacement, VRW=angle of right-hand screw displacement.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 a to 1 d show part of single-flighted self-cleaning screwprofiles according to the prior art in cross-section

FIGS. 2 a, 2 b, 2 d and 2 e show example cross-sections of partialprofiles of screw profiles according to the invention with a reduced tipangle.

FIG. 2 c shows a design according to the disclosure in U.S. Pat. No.3,900,187 in which, at a given tip angle and a given ratio RA, there areno further possible variations.

FIG. 2 f to 2 j show for illustrative purposes a longitudinal view ofscrews,

FIG. 3 a to 3 d show examples of a partial profile of a generating screwprofile,

FIG. 4 a to 4 d show partial profiles of screw profiles according to theinvention without kinks

FIG. 5 a to 5 c show the production of pairs of self-cleaning screws bythe displacement of screw profiles according to the invention in thedirection of the x axis,

FIG. 6 to 6 d shows a particular embodiment of screw elements,

FIG. 7 a to 7 d show examples of profiles of screw elements according tothe invention with gaps,

FIG. 8 a to 8 d show eccentric profiles obtained by designing a screwprofile with gaps and then displacing the profiles within the gaps.

FIG. 9 a shows a longitudinal view of a conveying thread according tothe invention,

FIG. 9 b shows a kneading element with seven kneading discs arranged onan axis at an offset angle of 60°.

DETAILED DESCRIPTION OF THE INVENTION

FIGS. 1 a to 1 d in each case show part of single-flighted self-cleaningscrew profiles according to the prior art in cross-section, as they aredescribed in Kohlgrüber.

The coordinate origin indicates the point of rotation. Circular arcs 1,2, 2′ and 1′ form one half of the screw profile. The other half isobtained by mirroring the profile shown at the horizontal straight linesthrough the point of rotation. The screw profile of the second screw,not shown, is obtained by displacing the shown and mirrored screwprofile by the amount A (normalized centreline distance) along thehorizontal straight lines through the point of rotation. The circulararc 1′ is additionally the generated circular arc related to thegenerating circular arc 1, as circular arc 2′ is the generated circulararc related to the generating circular arc 2.

The centre points of the circular arcs are illustrated by small circles.The centre points of the circular arcs are connected by thin, continuouslines both with the starting point and with the end point of theassociated circular arc. Outside the screw profile, the outer screwradius is indicated by a thin, dashed line.

In FIGS. 1 a to 1 d, the normalized outer radius RA is graduallyenlarged, changing from a flat cut profile (FIG. 1 a) to a deep-cutprofile (FIG. 1 d).

The circular arc “1” in each case represents half of the screw tip andthe associated angle α₁ the half tip angle. In a design according toKohlgrüber, α₁ has the magnitude

$\frac{\pi}{2} - {\arccos \left( \frac{a}{2 \cdot {ra}} \right)}$

and the sum of the tip angles of the two screws amounts to

${2\pi} - {4{{\arccos \left( \frac{a}{ra} \right)}.}}$

In FIG. 1 c, the half tip angle amounts for example to 52.5 degrees; thesum of all the tip angles of both screws amounts to 210 degrees.

FIGS. 2 a, 2 b, 2 d and 2 e show example cross-sections of partialprofiles of screw profiles according to the invention with a reduced tipangle. Axially symmetrical complete profiles may be produced bymirroring the partial profiles shown at the horizontal straight linesthrough the point of rotation. The profile of the second screw may beproduced from that shown and mirrored by displacement along thehorizontal straight line through the centre point of rotation by anamount A. The circular arc 1′ is additionally the generated circular arcrelated to the generating circular arc 1, as the circular arcs 2′ and 3′are the generated circular arcs related to the generating circular arcs2 and 3.

The ratio RA has the value 0.63 as in FIG. 1 c, but the half tip angleα₁ was reduced to 15 degrees and the sum of all the tip angles of bothscrews correspondingly to 60 degrees. In FIGS. 2 a to 2 d, the radiusR₂=0 was selected in each case, such that an edge arises at thetransition between the screw tip and flank and the radius R_(2′)corresponding to R₂ assumes the maximum value R_(2′)=A. In the figures,the radius R₃ was varied between R₃=0.9135 (FIG. 2 a) and R₃=0.5523(FIG. 2 e). FIGS. 2 a to 2 e illustrate the possibilities for variationof the design according to the invention, which at a given tip angle anda given ratio RA, encompasses a design ranging from angular profiles(FIG. 2 a) to highly rounded profiles (FIG. 2 e). FIG. 2 c shows adesign according to the disclosure in U.S. Pat. No. 3,900,187 in which,at a given tip angle and a given ratio RA, there are no further possiblevariations. According to U.S. Pat. No. 3,900,187, the centre point ofthe flank circle which adjoins the screw tip (in FIG. 2 the circle “3”)lies on the perpendicular to the axis of symmetry of the profile, whichaxis of symmetry passes through the point of rotation (in FIG. 2 the ycoordinate axis). It will be noted how the nip angle between thetangents to the flank of the screw profile and to the barrel circle atthe transition point between screw tip and flank is influenced in FIGS.2 a to 2 e by the variation of R₃. Depending on process requirements, itis possible according to the invention to select between, on the onehand, a design with a very acute angle, as shown in FIG. 2 e, whichleads to rotation of the screw to a great extent drawing the material tobe processed into the gap between screw tip and housing wall, and, onthe other hand, designs with a larger nip angle in which the screw flankpushes the material in front of it to a greater extent.

FIGS. 2 f to 2 j show for illustrative purposes a longitudinal view ofscrews, which are made up of screw profiles 2 a to 2 e and areconstructed as a conveying thread.

FIGS. 3 a to 3 d show examples of a partial profile of a generatingscrew profile according to the invention with RA=0.58, the half tipangle α₁ varying from 47.6 degrees in FIGS. 3 a to 11.9 degrees in FIG.3 d. The transition from the screw tip to the flank is rounded in FIGS.3 a to 3 d by the selection of a radius R₂ other than zero. Radius R₃,on the other hand, was selected with R₃ at most equalling A, such thatthe corresponding radius R_(3′) disappears. This gives rise to an edgein the flank of the screw profile, which edge does however rotate at agreater distance from the barrel. FIGS. 3 a to 3 d show the possiblevariations in the conspicuousness of this edge: when a small tip angleα₁ and a large flank angle α₃ are selected, the edge is highlyconspicuous (FIG. 3 d), when a large tip angle α₁ and small flank angleα₃ are selected, it is only slightly conspicuous (FIG. 3 a).

FIGS. 4 a to 4 d show partial profiles of screw profiles according tothe invention without kinks In a manner similar to FIGS. 3 a to 3 d, thehalf tip angle α₁ was varied from 47.6 degrees in FIGS. 4 a to 11.9degrees in FIG. 4 d. The flank radii were selected as R₂=0.125 andR₃=0.75. The variation in tip width here produces a range of screwprofiles, which extends from profiles with a pronounced high-shear andlow-shear zone (FIG. 4 a) to profiles with a homogeneous distribution ofshear rate over the circumference of the profile (FIG. 4 d).

FIGS. 5 a to 5 c show by way of example the production of pairs ofself-cleaning screws by the displacement of screw profiles according tothe invention in the direction of the x axis. If the points of rotationof the two screws lie on the x-axis, the profiles may, while maintainingfixed points of rotation, be displaced within the barrel radius in thedirection of the x-axis. In this manner, complete mutual cleaning of thescrew profiles is maintained FIG. 5 a shows the starting profile with ahalf tip angle α₁ of 23.8°, and flank radii of R₂=0.25, and R₃=0.75. InFIG. 5 b, the profile is displaced in the negative x axis direction. Thebarrel bore is now cleaned with a greatly enlarged gap. As a result, thezone of high shear between screw tip and barrel is no longer inexistence. FIG. 5 c shows the maximum possible displacement of theprofile. The screw root or groove of the original profile has nowassumed the function of the screw tip and vice versa.

One particular embodiment of screw elements according to the inventionis illustrated by way of example in FIGS. 6 a to 6 d.

The screw profile shown may be generated from the partial profile shownin FIG. 5 a by mirroring at the horizontal straight lines through thepoint of rotation. The profile of the second screw corresponds to theprofile of the first screw displaced by the amount A along the axis ofsymmetry. This embodiment is characterized in that the barrel bores areconstructed with a normalized radius RG=0.63, which is larger than theouter radius RA=0.58 of the screw profiles. The screw profiles aredisplaced in pairs relative to the centre points of the barrel bores.The points of rotation (shown by small circles), however, remain in thecentres of the barrel bores. Eccentrically rotating screw elements areproduced in this manner. Displacement within the barrel bores may beselected at will. FIGS. 6 a to 6 d show by way of example fourdisplacements for one and the same screw profile, in which in each casea different radius of the profile contour cleans the barrel bore.

The text has hitherto related to fully self-wiping screw profiles. Inmachines constructed industrially, it is, however, necessary to deviatefrom the fully self-wiping geometry to such an extent that preciselydefined gaps are maintained during cleaning. This is necessary in orderto prevent metallic “fretting”, to compensate for manufacturingtolerances and to avoid excessive energy dissipation in the gaps. Thereare various possible strategies for producing uniform gaps. The mostwidespread is the production of gaps which are equidistant over alongitudinal section through the machine. The procedure for generatingthe corresponding screw profiles was shown in Kohlgrüber on pages 103 etseq.

FIGS. 7 a to 7 d show examples of profiles of screw elements accordingto the invention with gaps (clearances). In FIG. 7 a, the gap S,normalized to the centreline distance, on mutual cleaning of the screwswas selected to be identical to the normalized gap D on cleaning of thebarrel. In FIG. 7 b, gap S is smaller than D and in FIGS. 7 c and 7 d Dis conversely smaller than S.

FIGS. 8 a to 8 d show that eccentric profiles according to the inventionmay also be obtained by designing a screw profile with gaps and thendisplacing the profiles within the gaps. The profiles of FIGS. 8 a to 8d are identical to the profile from FIG. 7 d. Displacement proceeds inrelation to a straight line through the points of rotation of the screwelements by an angle of 0° in FIG. 8 a, an angle of 60° in FIG. 8 b, anangle of 120° in FIG. 8 c and an angle of 180° in FIG. 8 d.

FIGS. 8 a to 8 d show examples in which both screws are displaced by thesame displacement vector. It is, in principle, also possible to displaceboth screws by a different vector within the clearances. In this case,profiles are obtained which clean one another with a gap which variesover one revolution of the screws.

As is known, the conveying action of a pair of profiles comes about bythe profiles being continuously helically rotated in the axialdirection. A conveying thread is obtained in this manner. FIG. 9 a showsby way of example a longitudinal view of a conveying thread according tothe invention.

Kneading elements with an elevated dispersing capacity relative to theconveying thread are obtained by arranging self-cleaning profileprismatic discs offset relative to one another on the axis. FIG. 9 bshows an example of a kneading element with seven kneading discs whichare arranged on the axis at an offset angle of 60°.

Without exception, the figures show symmetrical screw profiles. It is,however, also possible to produce asymmetric screw profiles. This isexplained in detail in PCT/EP2009/003549. For example the halves of ascrew profile shown in FIGS. 2 a and 2 b may be combined to form anasymmetric screw profile, for example by mirroring the profile shown inFIG. 2 b at the x axis and the mirrored part completing the missingportion of the profile in FIG. 2 a.

In the figures, at most 12 circular arcs are used to describe agenerating or a generated screw profile. The processes according to theinvention are, however, in no way limited to at most 12 circular arcs.Instead, as many circular arcs as desired may be used to generate screwprofiles. In particular, screw profiles which are not made up ofcircular arcs and are thus not self-cleaning may consequently beapproximated with a desired level of accuracy by a sufficiently largenumber of circular arcs.

The longitudinal section profile may be calculated from a screw'scross-sectional profile. Each circular arc of a screw profile ispreferably used in order to calculate a part of the longitudinal sectionbelonging to said circular arc by means of an explicit function.

The distance s of a point of a circular arc of a screw profile from theaxis of rotation is calculated in a first step by determining the pointof intersection (Sx, Sy) of a straight line g, characterized in thatsaid straight line lies in the plane of the screw profile, passesthrough the point of rotation of the screw profile and the orientationof the straight line is defined by the angle φ, with a circular arc kb,characterized by its radius r and the location of its centre point (Mx,My). In a second step, the distance s of the point of intersection (Sx,Sy) from the point of rotation of the screw profile is calculated. Thecalculation of a point of intersection of a straight line with acircular arc may be represented by an explicit function. The sameapplies to the distance calculation. s=s(φ, r, Mx, My) accordinglyapplies for the distance. Given a known pitch t of a screw element, theangle φ may be converted by φ/2π*t into an axial position z_ax, suchthat s=s(z_ax, r, Mx, My)=s(φ/2π*t, r, Mx, My) applies for the distance.The function s(z_ax, r, Mx, My) describes the desired longitudinalsection for a circular arc of the screw profile.

A process with which the profiles of screw elements according to theinvention may be designed is described below by way of example.

The process for generating closely intermeshing, self-cleaning,co-rotating screw profiles with a selectable centreline distance abetween the axes of rotation of a generating and a generated screwprofile is characterized in that the generating screw profile is formedfrom n circular arcs and the generated screw profile is formed from n′circular arcs, wherein

-   -   the generating screw profile and the generated screw profile lie        in one plane,    -   the axis of rotation of the generating screw profile and the        axis of rotation of the generated screw profile are in each case        perpendicular to said plane of the screw profiles, the point of        intersection of the axis of rotation of the generating screw        profile with said plane being designated as the point of        rotation of the generating screw profile and the point of        intersection of the axis of rotation of the generated screw        profile with said plane being designated as the point of        rotation of the generated screw profile,    -   the number of circular arcs n of the generating screw profile is        selected, n being an integer which is greater than or equal to        1,    -   an outer radius ra of the generating screw profile is selected,        wherein ra may assume a value which is greater than 0 (ra>0) and        less than or equal to the centreline distance (ra≦a),    -   a core radius ri of the generating screw profile is selected,        wherein ri may assume a value which is greater than or equal to        0 (ri≧0) and less than or equal to ra (ri≦ra),    -   the circular arcs of the generating screw profile are arranged        clockwise or counterclockwise around the axis of rotation of the        generating screw profile in accordance with the following rules        of arrangement, such that:        -   all the circular arcs of the generating screw profile merge            tangentially into one another in such a way that a            continuous, convex screw profile is obtained, wherein a            circular arc, whose radius is equal to 0, is preferably            treated as a circular arc whose radius is equal to eps,            wherein eps is a very small positive real number which tends            towards 0 (eps<<1, eps→0),        -   each of the circular arcs of the generating screw profile            lies within or at the limits of a circular ring with the            outer radius ra and the core radius ri, the centre point of            which lies on the point of rotation of the generating screw            profile,        -   at least one of the circular arcs of the generating screw            profile touches the outer radius ra of the generating screw            profile,        -   at least one of the circular arcs of the generating screw            profile touches the core radius ri of the generating screw            profile,    -   the magnitude of a first circular arc of the generating screw        profile, which is established by an angle α₁ and a radius r₁, is        selected such that the angle α₁ in radians is greater than or        equal to 0 and less than or equal to 2π, wherein π should be        taken to mean the circle constant (π≈3.14159), and the radius r₁        is greater than or equal to 0 and less than or equal to the        centreline distance a, and the position of this first circular        arc of the generating screw profile, which is obtained by the        positioning of two different points of this first circular arc,        is established in accordance with said rules of arrangement,        wherein a first point to be positioned of this first circular        arc is preferably a starting point belonging to this first        circular arc and wherein a second point to be positioned of this        first circular arc is preferably the centre point belonging to        this first circular arc,    -   the magnitudes of a further n-2 circular arcs of the generating        screw profile, which are established by the angles α₂, . . . ,        α_(n-1) and the radii r₂, . . . , r_(n-1), are selected such        that the angle α₂, . . . , α_(n-1) in radians is greater than or        equal to 0 and less than or equal to 2π and the radii r₂, . . .        , r_(n-1) are greater than or equal to 0 and less than or equal        to the centreline distance a, and the positions of these further        n-2 circular arcs of the generating screw profile are        established in accordance with said rules of arrangement,    -   the magnitude of a last circular arc of the generating screw        profile, which is established by an angle α_(n) and a radius        r_(n), is determined in that the sum of the n angles of the n        circular arcs of the generating screw profile in radians is        equal to 2π, wherein the angle _(—) _(n) in radians is greater        than or equal to 0 and less than or equal to 2π, and the radius        r_(n) closes the generating screw profile, wherein the radius        r_(n) is greater than or equal to 0 and less than or equal to        the centreline distance a, and the position of this last        circular arc of the generating screw profile is established in        accordance with said rules of arrangement,    -   the n′ circular arcs of the generated screw profile are obtained        from the n circular arcs of the generating screw profile in that        -   the number of circular arcs n′ of the generated screw            profile is equal to the number of circular arcs n of the            generating screw profile, n′ being an integer,        -   the outer radius ra′ of the generated screw profile is equal            to the difference of the centreline distance minus the core            radius ri of the generating screw profile (ra′=a−ri),        -   the core radius ri′ of the generated screw profile is equal            to the difference of the centreline distance minus the outer            radius ra of the generating screw profile (ri′=a−ra),        -   the angle α_(i′) of the i′th circular arc of the generated            screw profile is equal to the angle α_(i) of the ith            circular arc of the generating screw profile, wherein i and            i′ are integers which pass jointly through all the values in            the range from 1 to the number of circular arcs n or n′            respectively (α_(1′)=α₁, . . . , α_(—) _(n′) =α_(n)),        -   the sum of the radius r_(i′) of the i′th circular arc of the            generated screw profile and of the radius r, of the ith            circular arc of the generating screw profile is equal to the            centreline distance a, wherein i and i′ are integers which            pass jointly through all the values in the range from 1 to            the number of circular arcs n or n′ respectively            (r_(1′)+r₁=a, . . . , r_(n′)+r_(n)=a),        -   the centre point of the i′th circular arc of the generated            screw profile is at a distance from the centre point of the            ith circular arc of the generating screw profile which is            equal to the centreline distance a, and the centre point of            the i′th circular arc of the generated screw profile is at a            distance from the point of rotation of the generated screw            profile which is equal to the distance of the centre point            of the ith circular arc of the generating screw profile from            the point of rotation of the generating screw profile, and            the connecting line between the centre point of the i′th            circular arc of the generated screw profile and the centre            point of the ith circular arc of the generating screw            profile is a line parallel to a connecting line between the            point of rotation of the generated screw profile and the            point of rotation of the generating screw profile, i and i′            being integers which pass jointly through all the values in            the range from 1 to the number of circular arcs n or n′            respectively (i′=i),        -   a starting point of the i′th circular arc of the generated            screw profile lies in a direction relative to the centre            point of the i′th arc of the generated screw profile which            is opposite to that direction which has a starting point of            the ith circular arc of the generating screw profile            relative to the centre point of the ith circular arc of the            generating screw profile, i and i′ being integers which pass            jointly through all the values in the range from 1 to the            number of circular arcs n or n′ respectively (i′=i).

According to the invention, the circular arcs of the generating andgenerated screw profile should be selected or adapted to one anothersuch that the sum of the tip angles of a generating and of a generatedscrew profile is less than

${2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}$

and in the case of axially symmetrical screw profiles none of the centrepoints of the flank circles lies on the perpendicular to the axis ofsymmetry of the profile, which axis of symmetry passes through the pointof rotation.

From the described process for producing smooth, closely intermeshing,self-cleaning and co-rotating screw profiles, it follows for thegenerated screw profile that

-   -   the generated screw profile is continuous,    -   the generated screw profile is convex,    -   each of the circular arcs of the generated screw profile merge        tangentially into the following circular arc of the generated        screw profile, wherein a circular arc, whose radius is equal to        0, is preferably treated as a circular arc whose radius is equal        to eps, wherein eps is a very small positive real number which        tends towards 0 (eps<<1, eps→0),    -   each of the circular arcs of the generated screw profile lies        within or at the limits of a circular ring with the outer radius        ra′ and the core radius ri′, the centre point of which lies on        the point of rotation of the generated screw profile,    -   at least one of the circular arcs of the generated screw profile        touches the outer radius ra′ of the generated screw profile,    -   at least one of the circular arcs of the generated screw profile        touches the core radius ri′ of the generated screw profile.

It additionally follows from the above-described process for producingsmooth, closely intermeshing, self-cleaning, co-rotating screw profilesthat only in the case in which the core radius ri of the generatingscrew profile is equal to the difference of the centreline distance aminus the outer radius ra of the generating screw profile (ri=a−ra) isthe outer radius ra′ of the generated screw profile equal to the outerradius ra of the generating screw profile and the core radius ri′ of thegenerated screw profile equal to the core radius ri of the generatingscrew profile.

If the generating screw profile has a circular arc with the radiusr_(i)=0, the screw profile has a kink at the position of the circulararc, the magnitude of which kink is characterized by the angle α_(i). Ifthe generated screw profile has a circular arc with the radius r_(i)=0,the screw profile has a kink at the position of the circular arc, themagnitude of which kink is characterized by the angle α_(i).

The above-described process for producing smooth, closely intermeshing,self-cleaning, co-rotating screw profiles is furthermore distinguishedin that it can be performed solely with a set square and pair ofcompasses. The tangential transition between the ith and the (i+1)thcircular arc of the generating screw profile is thus designed bydescribing a circle with the radius r_(i+1) about the end point of theith circular arc, and the point of intersection, located closer to thepoint of rotation of the generating screw profile, of this circle withthe straight line which is defined by the centre point and the end pointof the ith circular arc is the centre point of the (i+1)th circular arc.In practice, instead of a set square and pair of compasses, computersoftware is used to design the screw profiles.

The screw profiles generated using the general process are independentof the number of flights z.

The generated screw profile may be different from the generating screwprofile. As a person skilled in the art will readily understand from theexplanations, the above-described method is suitable in particular forgenerating transition elements between screw elements with differentnumbers of flights. On the basis of a z-flighted screw profile, it ispossible to change the generating and the generated screw profiles stepby step such that a screw profile is ultimately obtained which has anumber of flights z′ different from z. It is in this respect admissibleto reduce or increase the number of circular arcs during the transition.

In the case of symmetrical profiles, the process may be simplified bydesigning only parts of the screw profiles and generating the missingparts from the designed parts by symmetry operations. This is describedin detail in PCT/EP2009/003549.

It is recommended that the process for producing screw profiles becarried out on a computer. The dimensions of the screw elements are thenpresent in a form in which they may be supplied to a CAD milling machinefor producing the screw elements.

1-15. (canceled)
 16. A screw elements for multi-screw extruderscomprising a plurality of screws co-rotating in pairs and being fullyself-wiping in pairs, each pair having a tip angle and one screw flight,with a centreline distance a and outer radius ra, wherein the sum of thetip angles of the pair of screws is less than${2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}$ and inthe case of axially symmetrical screw profiles none of the centre pointsof the flank circles lies on the perpendicular to the axis of symmetryof the profile, which axis of symmetry passes through the point ofrotation.
 17. The screw elements according to claim 16, wherein in eachcase one cross-sectional profile is composed of at least five circulararcs with a radius greater than or equal to zero and less than or equalto a distance a, the circular arcs merging tangentially into one anotherat their end points.
 18. The screw elements according to claim 16,wherein the screw elements are mixing elements or conveying elements.19. The screw elements according to claim 16, wherein the screw elementsare kneading elements.
 20. The screw elements according to claim 16,wherein the screw elements display clearances between screw elements anda barrel and/or between neighbouring screw elements.
 21. Screw elementsaccording to claim 16, wherein the screw elements each have an outerscrew radius normalized to the centreline distance, lies in the rangefrom 0.52 to 0.66.
 22. A method of using screw elements for multi-screwextruders, comprising the steps of providing a plurality of screwsco-rotating in pairs and being fully self-wiping in pairs, each pairhaving a tip angle and one screw flight, with a centreline distance aand outer radius ra, wherein the sum of the tip angles of the pair ofscrews is less than${2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}$ and inthe case of axially symmetrical screw profiles none of the centre pointsof the flank circles lies on the perpendicular to the axis of symmetryof the profile, which axis of symmetry passes through the point ofrotation.
 23. The method according to claim 22, wherein the screwelements clean one another in pairs with a constant gap over theirentire circumference.
 24. The method according to claim 22, wherein thescrew elements clean one another in pairs with a gap which is notconstant over the entire circumference.
 25. The method according toclaim 22, wherein the profiles of the screw elements are displaced inpairs relative to the point of rotation located centrally in the barrelbore.
 26. A process for extruding plastic compositions in a twin-screwor multi-screw extruder using screw elements, comprising the steps ofproviding a plurality of screws co-rotating in pairs and being fullyself-wiping in pairs, each pair having a tip angle and one screw flight,wherein the sum of the tip angles of a generating and of a generatedscrew profile is less than${2\pi} - {4{\arccos \left( \frac{a}{2 \cdot {ra}} \right)}}$ and inthe case of axially symmetrical screw profiles none of the centre pointsof the flank circles lies on the perpendicular to the axis of symmetryof the profile, which axis of symmetry passes through the point ofrotation.
 27. The process according to claim 26, wherein the plasticcompositions are thermoplastics or elastomers.
 28. The process accordingto claim 27, wherein the thermoplastics used are polycarbonate,polyamide or polyester or a blend of at least two thermoplastics. 29.The process according to claim 27, wherein the thermoplastics used arepolybutylene terephthalate and polyethylene terephthalate, polyether,thermoplastic polyurethane, polyacetal or fluoropolymer or a blend of atleast two of the stated thermoplastics
 30. The process according toclaim 27, wherein the thermoplastics used are polyvinylidene fluoride,polyether sulphones, polyolefin or a blend of at least two of the statedthermoplastics.
 31. The process according to claim 27, wherein thethermoplastics used are polyethylene and polypropylene, polyimide,polyacrylate or a blend of at least two of the stated thermoplastics.32. The process according to claim 27, wherein the thermoplastics usedare poly(methyl)methacrylate, polyphenylene oxide, polyphenylenesulphide, polyether ketone, polyarylether ketone, styrene polymers or ablend of at least two of the stated thermoplastics.
 33. The processaccording to claim 32, wherein the styrene polymers are polystyrene,styrene copolymers or a blend of at least two of the statedthermoplastics.
 34. The process according to claim 33, wherein thestyrene copolymers are styrene-acrylonitrile copolymer,acrylonitrile-butadiene-styrene block copolymers, polyvinyl chloride ora blend of at least two of the stated thermoplastics.
 35. The processaccording to claim 27, wherein the elastomers used are styrene-butadienerubber, natural rubber, butadiene rubber, isoprene rubber,ethylene-propylenediene rubber, ethylene-propylene rubber,butadiene-acrylonitrile rubber, hydrogenated nitrile rubber, butylrubber, halobutyl rubber, chloroprene rubber, ethylene-vinyl acetaterubber, polyurethane rubber, thermoplastic polyurethane, gutta percha,acrylate rubber, fluororubber, silicone rubber, sulphide rubber,chlorosulphonyl-polyethylene rubber or a combination of at least two ofthe stated elastomers.
 36. The process according to claim 27, wherein afiller or reinforcing materials or polymer additives or organic orinorganic pigments or mixtures thereof are added to the plasticscompositions.